Theorem pm2.68dc 826

Description: Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 699 and one half of dfor2dc 827. (Contributed by Jim Kingdon, 27-Mar-2018.)

Assertion

Ref	Expression
pm2.68dc	⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))

Proof of Theorem pm2.68dc

Step	Hyp	Ref	Expression
1		jarl 616	. 2 ⊢ (((𝜑 → 𝜓) → 𝜓) → (¬ 𝜑 → 𝜓))
2		pm2.54dc 823	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))
3	1, 2	syl5 32	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  dfor2dc  827